During this lesson, students will engage in activity where they will
investigate relationships of sides and angles in a triangle. While
students are engaging in the activity, the teacher's role will be that
of facilitator. When students complete the activity, the teacher will
use the questions on the activity sheet to lead discussion about the
important concepts. The lesson flows well when students begin with the
activity for the triangle inequality and move to the activity for the
inequalities for sides and angles of a triangle.
Provide students with the The Triangle Inequality Activity Sheet, a ruler, and 8‑10 pieces of thin pasta. Students should
work individually on the first five questions on the activity sheet,
and they should work with a partner on the last three questions.
Some students may not understand what it means to have three sides
that cannot make a triangle (Question #2 in the activity sheet). In an attempt to create a non‑triangle,
students may arrange three sides that form a triangle in such a way
that they appear to form a non-triangle. It is important to stress that
it must be impossible to form a triangle regardless of how the three
pieces are arranged.
When students describe the relationship between the sum of the
measures of the small and medium sides and the measure of the large
side of the triangles and the non-triangles, some students will
describe the sum of the measures of the small and medium sides as "far
away from", "close to", or "different from" the measure of the large
side. As students begin to make their conjectures in Question 5,
encourage them to refine their ideas.
After students have had time to work on Questions 6‑8 with their
partner, use these questions to focus class discussion about the
triangle inequality. Ask students to share their conjectures, and most
importantly why they think their conjecture would work for any
triangle. Make certain that students realize that testing a few
triangles is not sufficient evidence that a conjecture will work for
every triangle. Encourage students to reason about the relationships
between the sum of measures of the small and medium sides and the
measure of the large side of the triangle. For example, ask students
why it is impossible to have a triangle where the sum of the measures
of the small and medium sides is less than the sum of the measure of
the large side.
When discussing Question 7 on the activity sheet, use pasta to
demonstrate why it is impossible for the sum of measures of the small
and medium sides to be equal to the measure of the large side of a
triangle. Break pasta into two equal parts. Let one part be the large
side of the triangle. Break the other part into two pieces (not
necessarily equal), and let those be the small and medium sides of the
triangle. The small and medium sides collapse onto the large side,
creating a line segment, not a triangle.
When discussing Question 8 on the activity sheet, describe how the
sum of the measures of any other pair of sides would have to be greater
than the measure of the remaining side, because the measure of at least
one of the sides in the pair would already be greater than the measure
of the remaining side before finding the sum. For example, the sum of
the measures of the large and small sides would have to be greater than
the measure of the medium side, because the large side is greater than
the medium side.
Conclude the triangle inequality part of the lesson by discussing
the significance of finding the sum of the measures of the small and
medium sides of the triangle. Since the triangle inequality states that
the sum of the measures of any two sides of a triangle will always be
greater than the measure of the remaining side, by determining that the
sum of the measures of the small and medium sides is greater than the
measure of the large side, one can be certain that the other
inequalities will hold.
To begin the inequalities for sides and angles of a triangle part of the lesson, provide students with the
Inequalities for Sides and Angles of a Triangle activity sheet and a protractor. Students should begin by working individually on the Activity Sheet.
While students are working on the activity sheet, circulate
around the room to monitor student progress. Students who have not used
a protractor on a regular basis can have difficulty measuring angles.
Remind students to consider whether or not their three angle
measurements are reasonable based on the sum of angles in a triangle.
Students may have difficulty understanding what it means for a side to
be opposite an angle in a triangle. It may be helpful to draw a
diagram, such as the one below.
Moreover, it is important for students to realize that although the
longest side of the triangle is opposite the greatest angle, a side
opposite an 82°‑angle in one triangle can be greater than the side
opposite a 115°‑angle in another triangle. For example, provide
students with a diagram such as the one below, and ask students if it
is possible for AC > DF, if m∠B = 82°, and m∠E = 115°.
[Yes, because all that is known for certain is that AC and DF
are greater than the measures of any other sides in their respective
triangles. There is no way to know how their lengths compare to the
side lengths of other triangles.]
In reasoning about Question 7 on the activity sheet, an acceptable
argument would be that if a scalene triangle had two congruent angles,
then those angles would have to be opposite congruent sides, but since
the scalene triangle has no congruent sides this is impossible.
Conclude the lesson by asking students to describe both the triangle
inequality and the inequalities for sides and angles of a triangle
using words and symbols.
- The sum of the measures of any pair of sides in a triangle will always be greater than the measure of the remaining side.
[s + m > l; s=shortest side, m=second shortest line, and l.]
longest side of a triangle will always be opposite the greatest angle
of the triangle, and the shortest side will always be opposite the
[Given ΔABC with m∠A > m∠B > m∠C, we know that BC > AC > AB.]